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Designs, Codes and Cryptography

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01-09-2014

The Ubiquity of the Orders of Fractional Semifields of Even Characteristic

Author: Vikram Jha

Published in: Designs, Codes and Cryptography | Issue 3/2014

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Abstract

To each non-square integer \(2^{2N+1}\ge 2^5\) there correspond semifields \(D\) of order of \(2^{2N+1}\) that contain \(\text{ GF}(4)\). Hence there exist affine planes for each non-square order \(2^{2N+1}\ge 2^{5}\) that contain subaffine planes of order \(2^2\). Moreover, there also exists semifields \(D_1\) and \(D_2\), with \(|D_1|= |D_2| =|D|\) such that \(D_1\) is commutative and \(D_2\) is non-commutative but neither \(D_1\) nor \(D_2\) contains \(\text{ GF}(4)\).

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The procedure fails for strict presemifields \((F,+, \bullet )\) whenever \(\mathcal F \) does not include the identity map on \(F\).

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Title: The Ubiquity of the Orders of Fractional Semifields of Even Characteristic
Author: Vikram Jha
Publication date: 01-09-2014
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 3/2014
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-013-9795-6

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